An operational model that analyzes the relationship between annual energy output and year-end stored energy of a reservoir system was developed and used for determining the optimal year-end water level of a multi-year regulating storage reservoir. The model accounts for the capacity benefit of a multi-annual storage plant and was validated using the case of the Longyangxia reservoir in the upper reaches of the Yellow River, China. The critical elements for determining the optimal year-end water level for multi-year regulating storage reservoirs were revealed through analysis of the impacts of runoff and year-start water level on year-end water level. Simulated operational results from the model were compared with practical operational results for the Longyangxia reservoir system from 1990–2012. A comparison shows that the operating mode for Longyangxia reservoir (after 2006) achieved significant improvement versus before 2006, in making the reservoir run at a higher water level and increasing benefits. In addition, our study indicates that the model can be effectively used for multi-year regulating storage reservoir operation.

INTRODUCTION

Regulation of discharge from a multi-year regulating storage reservoir requires full use of its utilizable capacity throughout the year. In other words, there is no need to discharge water from a multi-year regulating storage reservoir to have its year-end water level drop to the dead water level (Labadie 2004; Li et al. 2013). To make up for water shortages in low flow years, water would not be discharged to result in the year-end water level of the multi-year regulating storage reservoir declining to the dead water level (Simonovic 1992; Wurbs 1993). It is only necessary to control the year-end water level somewhere between the normal water level and dead water level, so that reservoir outflow simultaneously satisfies actual water demands and other complicated constraints. This water level is referred to as the optimal year-end water level of a multi-year regulating storage reservoir (Tian et al. 1999; Guo et al. 2004; Wang et al. 2005; Zou & He 2006; Yuan & Wang 2012; Zhang et al. 2012; Xie et al. 2014). For a multi-year regulating storage reservoir, the optimal year-end water level is a determination that not only concerns water usage throughout a year, but also exerts an influence on water usage in subsequent years. The optimal year-end water level will influence its own operating income, but will also have a great impact on the operation of other hydroelectric power stations that are related to the multi-year regulating storage reservoir in terms of water and electric power (Tian et al. 1999). In recent years, with the large-scale and rapid development of water resources and hydropower engineering in China, the number of multi-year regulating storage reservoirs has constantly increased, and a deeper understanding of the compensation effects of multi-year regulating storage reservoirs on the power system and water dispatching system has been gained (Yuan & Wang 2012; Zhang et al. 2012; Xie et al. 2014). Determination of the optimal year-end water level for a multi-year regulating storage reservoir, therefore, has become one of the most complicated issues in water resources management because it typically involves trade-offs.

There are currently two methods for determining the optimal year-end water level for multi-year regulating storage reservoirs. The first is to obtain the optimal year-end water level by establishing the functional relationship between the year-end water level and related factors (Tian et al. 1999; Wang et al. 2005; Xie et al. 2014). The second is to establish an optimal operation model that incorporates the year-end water level or year-end stored energy of a multi-year regulating storage reservoir, and solve the model through optimization algorithms to obtain the optimal year-end water level that satisfies all constraints (Guo et al. 2004; Zou & He 2006; Yuan & Wang 2012; Zhang et al. 2012). Due to such disadvantages as small sample series, uncertainty of the function type and relatively large prediction error for the related factors, the first approach may be difficult when applied to actual situations (Zhang et al. 2012). The second method is more commonly used, but there is little research into the conditions that must be met for different optimal operation models to obtain the optimal year-end water level. This paper establishes an operational model for determining the optimal year-end water level of a multi-year regulating storage reservoir and conducts in-depth analysis of the Longyangxia reservoir system in the upper reaches of the Yellow River. It aims to find the essential elements that affect the optimal year-end water level for a multi-year regulating storage reservoir so as to provide theoretical support for pinpointing the optimal year-end water level of Longyangxia reservoir.

OPTIMIZATION MODEL

The energy stored by a multi-year regulating storage reservoir system is comprised of the annual energy output of the hydro plant and the year-end stored energy of the reservoir (Zou & He 2006). When the annual energy output is increased, the year-end stored energy of the reservoir will decrease and consequently the water used for power generation in the following years will also decrease. Conversely, when the year-end stored energy is increased, the water used for power generation throughout the year will decrease and the annual energy output will also decrease. Therefore, the key to determining the optimal year-end water level for a multi-year regulating storage reservoir system is to reconcile the conflict between the annual energy output and year-end stored energy. Considering that electric power systems are usually short of not only generating capacity, but also peak-load regulating capacity during low flow periods, we have developed an optimal operational model by maximizing the total energy of the multi-year regulating storage reservoir system while taking into account the requirement for annual guaranteed output of the hydro plant.

Objective function

The objective function of the aforementioned mathematical model is expressed as follows: 
formula
1
where E, and respectively represent the total energy, annual energy output and year-end stored energy of the multi-year regulating storage reservoir system (108 kW h) (Zou & He 2006; Yuan & Wang 2012; Wang et al. 2015), where 
formula
2
and 
formula
3
is the length of the scheduling period. Usually the scheduling period is assumed to be one year and the dispatch interval is set as a month (i.e. ). t denotes the period number, ; (h) is the time interval of the -th period; (m3/s) is the water discharge of the hydro plant in the -th period; (m3/kW·h) is the water consumption rate for power generation of the hydro plant in the -th period. Usually is obtained by means of the relation curve between (m) and through (Ma et al. 2011), where is the mean net head of the hydro plant in the -th period. (MW) is the annual guaranteed output of the hydro plant. The purpose of Equation (1) is to maximize the total energy of the multi-year regulating storage reservoir system while meeting, when possible, the requirement of (Huang et al. 2009; Wang et al. 2015). (108 m3) denotes the available water storage volume corresponding to the year-end water level of the reservoir; (m3/kW h) is the water consumption rate corresponding to for power generation of the hydro plant, and is obtained by means of the relation curve between and through (m), where is the mean net head corresponding to . In other words, a new concept, electrical equivalent of the reservoir (i.e. the energy for power generation corresponding to ), is adopted as the year-end stored energy of the multi-year regulating storage reservoir in Equation (3). This method has the advantage of using simple principles and calculations, such as and a being used as model parameters with , and when is set to 1,000; a usually takes the value of 1 or 2, and when a is set to 1; is a discrete variable ranging from 0–1, and is defined as follows: 
formula
4

Constraints

The above-mentioned objective function is subjected to the following equality and inequality constraints.

1. Water dynamic balance equilibrium

where (108 m3) is the reservoir storage volume at the beginning of the -th period; (108 m3) denotes the reservoir storage volume at the end of the -th period; , (m3/s) are the natural runoff and abandoned flow of the reservoir in the -th period, respectively.

2. Water level constraints

 
formula
6
where (m) denotes the dead water level of the reservoir or the minimum water level for comprehensive utility of the reservoir in the -th period; (m) is the upstream water level of the reservoir in the -th period; (m) represents the maximum water level permissible of the reservoir in the -th period, such as normal water level or flood control level (Ouyang et al. 2014).

3. Year-start water level limit

where (m) is the initial water level of the scheduling period of the reservoir; (m) is the upstream water level of the reservoir at the beginning of the first period (Karamouz et al. 2014).

4. Water release constraints

where (m3/s) denotes the minimum water discharge for comprehensive utility of the reservoir in the -th period; (m3/s) is the maximum water discharge permissible of the reservoir in the -th period; (m3/s) represents the minimum water discharge permissible of the water turbines of the hydro plant in the -th period; (m3/s) denotes the maximum flow capacity of the water turbines of the hydro plant in the -th period.

5. Power output upper and lower limits

 
formula
10

where (MW) denotes the minimum permissible output of the hydro plant in the -th period; (MW) is the capacity available or expected output of the hydro plant in the -th period.

6. Non-negativity conditions

All the aforementioned variables are larger than or equal to zero.

Solving the algorithm

Under the condition of satisfying all the above constraints, the total energy, annual energy output, and year-end stored energy corresponding to the optimal year-end water level for a multi-year regulating storage reservoir can be obtained by resolving the above-mentioned objective function with optimization algorithms and a given runoff process. In this paper, the process of optimization was used with the progressive optimality algorithm (POA), where POA is a traditional method that is widely used to achieve optimal operation of hydropower energy systems (Lucas & Perera 1985; Nanda et al. 1986; Huang et al. 2009).

SYSTEM CONSIDERED

The reservoir system, Longyangxia, used as a case study is in the upper reaches of the Yellow River in China. Longyangxia is not only the largest reservoir but is also the only multi-year regulating storage reservoir along the Yellow River. Its total storage capacity is 247 × 108 m3, which is basically equivalent to the mean annual runoff in the upper reaches of the Yellow River. As a result, the reservoir has the capacity to store plenty of water. The Longyangxia reservoir system was designed for multiple purposes including water supply, power generation, flood control, and ecological benefit. The operations of this multi-purpose project are monitored by the Yellow River Conservancy Commission. With the total installed capacity being 1,280 MW and designed guaranteed output being 589.8 MW, Longyangxia hydropower plant is the primary power source for the Northwest China Power Grid and plays a significant role in maintaining the stable operation of the power system. The main operation characteristics of Longyangxia hydropower energy system are listed in Table 1.

Table 1

Characteristics of Longyangxia hydropower energy system

ParameterValueParameterValue
Normal water level (m) 2,600 Total storage capacity (108 m3247 
Dead water level (m) 2,530 Utilizable capacity (108 m3193.5 
Flood control level (m) 2,594 Annual mean flow (m3/s) 650 
Installed capacity (MW) 1,280 Designed guaranteed output (MW) 589.8 
Power plant discharge range (m3/s) [130, 1,240] Reservoir discharge range (m3/s) [130, 6,000] 
ParameterValueParameterValue
Normal water level (m) 2,600 Total storage capacity (108 m3247 
Dead water level (m) 2,530 Utilizable capacity (108 m3193.5 
Flood control level (m) 2,594 Annual mean flow (m3/s) 650 
Installed capacity (MW) 1,280 Designed guaranteed output (MW) 589.8 
Power plant discharge range (m3/s) [130, 1,240] Reservoir discharge range (m3/s) [130, 6,000] 

ANALYSIS OF FACTORS

Many factors in theory have an influence on the year-end water level for a multi-year regulating storage reservoir but they generally belong to one of three categories: time, space, or energy factors. Time factors are related to the current and future status of the reservoir, such as year-start water level and runoff. Space factors include the conditions of other reservoirs closely related to the multi-year regulating storage reservoir, especially the water levels and discharges of the reservoirs with a large storage capacity. Energy factors primarily refer to the load demands of power systems as well as the requirements of multiple users for water. To confirm the correlation between all influencing factors and the year-end water level is very hard, due to extremely complicated relationships among the aforesaid factors. An impractical pursuit of precision and comprehensiveness will inevitably give rise to difficulties in analysis of the results of the presented model, and lose such advantages as being simple in calculation and easy to understand (Zhang et al. 2012). From the model described above, we find that runoff and year-start water level are the vitally related factors that have a great influence on the results, under the precondition of knowing the constraints and operational parameters of the reservoir system.

POA calculations were carried out with different runoff processes and different year-start water levels. The effects of runoff and year-start water level on year-end water level were analyzed to examine the impact of changing rules on the optimal year-end water level for Longyangxia reservoir. During the calculations, the scheduling period is assumed to be one calendar year, and the dispatch interval is set as a month. In addition, the annual guaranteed output in Equation (2) takes the designed guaranteed power output limit value of 589.8 MW. If the power output of Longyangxia hydro plant in any month fails to meet the designed guaranteed power output limit, we reset the value of according to the principle of maximizing minimum power output, namely the annual guaranteed output limit value, and will be set as large as possible until the power output of the hydro plant at any time does not satisfy it.

Impact of runoff

This section shows the results for modeling the impact of runoff on year-end water level. The year-start water level of Longyangxia reservoir was set to 2,570 m and the monthly distribution proportions of the annual mean flow for different supposed frequencies are one and the same, as given in Table 2. The annual mean flows of Longyangxia reservoir for different supposed frequencies are listed in Table 3. The simulated calculation results of the presented model are shown in Figure 1.
Table 2

Monthly distribution proportions of annual mean flow

Month Jan Feb Mar Apr May Jun 
Proportion 0.29 0.32 0.41 0.55 0.87 1.38 
Month Jul Aug Sep Oct Nov Dec 
Proportion 2.02 1.67 1.87 1.51 0.74 0.35 
Month Jan Feb Mar Apr May Jun 
Proportion 0.29 0.32 0.41 0.55 0.87 1.38 
Month Jul Aug Sep Oct Nov Dec 
Proportion 2.02 1.67 1.87 1.51 0.74 0.35 
Table 3

Annual mean flows of different supposed frequencies

Frequency 5% 10% 20% 30% 40% 50% 
Annual mean flow (m3/s) 980 873 761 691 639 596 
Frequency 60% 70% 80% 90% 95%  
Annual mean flow (m3/s) 557 522 487 449 426  
Frequency 5% 10% 20% 30% 40% 50% 
Annual mean flow (m3/s) 980 873 761 691 639 596 
Frequency 60% 70% 80% 90% 95%  
Annual mean flow (m3/s) 557 522 487 449 426  
Figure 1

Results of the model with different runoffs.

Figure 1

Results of the model with different runoffs.

Figure 1(a) shows the following:

  1. The year-end water level of Longyangxia reservoir in high flow years (the frequency of annual mean flow is lower than 30%, i.e. ) is approximately 2,585 m and no lower than 2,570 m, allowing abundant water to be stored. In normal flow years (40% ≤ Fw ≤ 60%), the year-end water level is controlled at approximately 2,570 m, which is basically equivalent to the year-start water level. Moreover, in low flow years , the year-end water level is no higher than 2,570 m and no lower than 2,548 m for the purpose of making up for the water shortage in the downstream reaches.

  2. If , the minimum power output of Longyangxia hydro plant is higher than 589.8 MW and meets the designed guaranteed power output limit. If , then it is impossible to fulfill the designed guaranteed power output during all the months of the year while simultaneously meeting other water-usage requirements within the Longyangxia reservoir system.

Figure 1(b) shows that, with the decrease of the annual mean flow for the Longyangxia hydropower energy system, the downward trend of the annual energy output gradually flattens while the year-end stored energy rapidly declines. That is because the water stored in the reservoir at the beginning of the year functions to make up for the water shortage for annual energy output so as to fulfill the annual guaranteed output.

Impact of year-start water level

For the analysis of the impact of decreasing year-start water level (from an elevation of 2,600 m down to 2,530 m) on year-end water level, the annual mean flow corresponding to 50% was selected. The simulated results are shown in Figure 2, from which the following is concluded:
  1. When the year-start water level of Longyangxia reservoir is not lower than 2,590 m, the year-end water level is maintained at 2,586 m. This is because the annual runoff volume is relatively lower than the year-start water storage of the reservoir. Total energy can be maximized while fulfilling the annual guaranteed output, and the relatively plentiful water stored in the reservoir at the beginning of the year is used to generate electricity. Under this scenario, a higher year-start water level equates to a low water consumption rate for power generation, which allows for more electricity generation from the water stored in the reservoir and maximizes the total energy of the year.

  2. With the year-start water level dropping from 2,590 m to 2,550 m, the year-end water level of Longyangxia reservoir and the corresponding variables such as the total energy, annual energy output and year-end stored energy suffer, and their changing trends are basically consistent with that of the year-start water level.

  3. When the year-start water level is lower than 2,550 m, the year-end water level and the corresponding year-end stored energy are higher when the year-start water level for Longyangxia reservoir is lower. Furthermore, the lower the year-start water level, the more significant the increases in the year-end water level and year-end stored energy. This relationship occurs because when maximizing the total energy by fulfilling the annual guaranteed output, the water consumption rate for power generation will be relatively high, and it is more likely for runoff to be stored in the reservoir. Water stored in the reservoir increases the water level and reduces the water consumption rate for power generation.

  4. When the year-start water level is higher than 2,570 m, the minimum power output of Longyangxia hydro plant can fulfill the designed guaranteed power output.

Figure 2

Results of the model with different year-start water levels.

Figure 2

Results of the model with different year-start water levels.

From the above conclusions, we can find that when the mean water consumption rate, , corresponding to the annual energy output, , equals the water consumption rate, , corresponding to the year-end stored energy, , the total energy, E, reaches a maximum value. If , the multi-year regulating storage reservoir system tends to store more water and decreases. Otherwise, the system tends to use more water to generate more electricity, as a result the reservoir water level decreases and increases. Under the condition of the year-start water level and runoff process given, the optimal year-end water level is the level that enables to equal . This is the nature of determining the optimal year-end water level for a multi-year regulating storage reservoir.

CASE STUDIES

Simulated calculations have been carried out for the actual runoffs of Longyangxia reservoir over the past 23 years (1990–2012) to verify the appropriateness of the proposed model. In the calculations, the actual year-start water level (2,574.62 m) of 1990 is used as the value for the initial water level of Longyangxia reservoir, and the optimal year-end water level is predicted to be the year-start water level of the next year. Using the model, both the optimal year-end water level and its corresponding annual energy output can be acquired. According to different methods adopted for determining the annual guaranteed output limit value of Longyangxia hydro plant, the comparative analyses between the simulated operation results and actual operation results are made in the following two ways.

First case

The annual guaranteed output, , in Equation (2) uses the designed guaranteed power output limit value of 589.8 MW. is reset according to the principle of maximizing minimum power output (i.e. the value of in Equation (2) will be reset as large as possible, until the power output is less than this value in one period) when the power output value of the Longyangxia hydro plant is less than 589.8 MW. The simulated results are shown in Figure 3, and lead to the following conclusions:
  1. Prior to 2006, the actual value of the average annual energy output of the Longyangxia hydro plant was 39.56 × 108 kW h, but the calculated value from the proposed model is 41.2 × 108 kW h, which is 1.64 × 108 kW h (4.15%) higher than the actual value. The actual value of the average annual minimum power output is 306 MW, but the average value of the annual minimum power output obtained from the model is 457 MW, which is 33.04% higher than the actual value. This indicates that prior to 2006, the model that fulfills the designed guaranteed power output or maximized minimum power output, generates better results than the actual operation results of the Longyangxia plant in terms of both power-generating benefits and capacity benefits, and especially a significant improvement for the annual minimum power output.

  2. After 2006, the actual value of the year-end water level for Longyangxia reservoir is above 2,580 m, maintaining a relatively high operating water level. Consequently, the water consumption rate for power generation is kept lower and the corresponding actual value of the average annual energy output is 60.17 × 108 kW h. However, in the proposed model, minimum power output of Longyangxia hydro plant is ensured to fulfill the designed guaranteed power output or maximized minimum power output, while the year-end water level drops to a low level and the water consumption rate for power generation increases. The obtained average annual energy output from the model is only 47.03 × 108 kW h, which is 13.14 × 108 kW h (21.84%) lower than the actual value. The actual average value of annual minimum power outputs is 443 MW, which is 71 MW (16.03%) less than the value of 514 MW from the model that fulfills the designed guaranteed power output or maximized minimum power output.

  3. The actual average value of annual energy output was 45.83 × 108 kW h from 1990 to 2012 for Longyangxia hydro plant, but the value from the model is 42.97 × 108 kW h, which is 2.86 × 108 kW h (6.24%) lower than the actual value. In 2012, the actual year-end water level of Longyangxia reservoir was 2,593.95 m and its corresponding year-end stored energy was 43.33 × 108 kW h, while the year-end water level from the proposed model was 2,584 m and the corresponding year-end stored energy was 32.63 × 108 kW h, which is 10.7 × 108 kW h (24.69%) lower than the actual value. The actual total energy of Longyangxia reservoir system during the 23 years from 1990–2012 was 1,097.42 (i.e. 45.83 × 23 + 43.33 = 1,097.42) × 108 kW h while the value from the model was 1,020.94 (i.e. 42.97 × 23 + 32.63 = 1,020.94) × 108 kW h or 6.97% less than the actual value. The actual average value of annual minimum power output was approximately 348 MW while the corresponding value from the model was 475 MW, an increase of 36.49% compared with the actual value.

Figure 3

Results from the model when the annual guaranteed output limit value is set to the designed guaranteed output or maximized minimum power output: (a) minimum power output and year-end water level; (b) annual energy output.

Figure 3

Results from the model when the annual guaranteed output limit value is set to the designed guaranteed output or maximized minimum power output: (a) minimum power output and year-end water level; (b) annual energy output.

A comparison of conclusion (1) and (2) shows that the operating mode of Longyangxia hydropower energy system seems to have been more reasonable and economical after 2006 versus before 2006. The foregoing conclusions, moreover, also indicate that if the annual guaranteed output for the considered system is set according to the principle of fulfilling the designed guaranteed power output or maximizing minimum power output in the developed model, the obtained solutions can maximize the regulating performance of the multi-year regulating storage reservoir as well as the installed capacity benefits of the hydro plant. However, this practice has a negative effect on the water head for power generation, thereby lowering the total generated energy.

Second case

The annual guaranteed output, , in Equation (2) takes the actual value of the minimum power output annually, as shown in Figure 3(a). The simulated results are shown in Figure 4, and several conclusions can be drawn:
  1. For the years prior to 2006, the results of the proposed model are significantly higher than the actual values, whether for the year-end water level or annual energy output. After 2006, however, the actual values tend to be consistent with the calculated results. It indicates that, restricted by the objective conditions, the water level of Longyangxia reservoir dropped to an overly low level before 2006, which resulted in operation at a relatively low water head for a long period of time. Prior to 2006, the average annual energy output of Longyangxia hydro plant was 52.35 × 108 kW h, which was 12.79 × 108 kW h (32.33%) higher than the actual value of 39.56 × 108 kW h. The operating mode prior to 2006 fails to bring into full play the power-generating benefits and installed capacity benefits of the considered system, and it is therefore an unreasonable and uneconomical operating mode. After 2006, however, both the actual annual year-end water level of Longyangxia reservoir and the calculated value from the proposed model are above 2,580 m, which leads to the reservoir operating at a relatively higher water level so as to reduce the water consumption rate for power generation. After 2006, the actual average annual energy output of Longyangxia hydro plant is 60.17 × 108 kW h, which is only 2.91% lower than the calculated value of 61.97 × 108 kW h. It is obvious that the operating mode of Longyangxia hydropower energy system experienced significant improvement after 2006 as compared to the pre-2006 period. Since 2006, it has been in a reasonable and economical operating mode.

  2. The average annual energy output calculated by the model from 1990–2012 was 55.27 × 108 kW h, which was an increase of 20.60% compared with the actual value of 45.83 × 108 kW h. The actual year-end water level for 2012 was 2,593.95 m and its corresponding year-end stored energy was 43.33 × 108 kW h, while the year-end water level obtained from the developed model was 2,588 m and its corresponding year-end stored energy was 36.75 × 108 kW h, a reduction of 15.19% compared with the actual value. The actual total energy was 1,097.42 (i.e. 45.83 × 23 + 43.33 = 1,097.42) × 108 kW h in the 23 years from 1990–2012, while the calculated value of the model was 1,308.04 (i.e. 55.27 × 23 + 36.75 = 1,307.96) × 108 kW h, an increase of 19.18% compared with the actual value. We conclude that the optimal year-end water level obtained by the model for multi-year regulating storage reservoirs is reasonable, effective, and applicable to production.

Figure 4

Results from the model when the annual guaranteed output limit value is set to the actual annual minimum power output.

Figure 4

Results from the model when the annual guaranteed output limit value is set to the actual annual minimum power output.

CONCLUSIONS

The optimization of year-end water level for a multi-year regulating storage reservoir has always been one of the most complicated issues in water resources management. An operational model for determining the optimal year-end water level was established in this paper by analyzing the relationship between annual energy output and year-end stored energy of the reservoir system, and was solved by means of POA. The Longyangxia reservoir in the upper reaches of the Yellow River was used as a case study to assess the impacts of runoff and year-start water level on year-end water level, which revealed the essential elements for determining the optimal year-end water level for a multi-year regulating storage reservoir. The conclusion is that an optimal year-end water level for a multi-year regulating storage reservoir is the one that enables (i.e. mean water consumption rate corresponding to annual energy output) to equal (i.e. water consumption rate corresponding to year-end stored energy). The case studies indicate that the operating modes of the Longyangxia reservoir system improved significantly after 2006 versus before 2006, and indicate that the operational model is reasonable and effective. In this paper, we analyze the essential elements for determining the optimal year-end water level for a multi-year regulating storage reservoir from the perspective of a single reservoir operation. Further study thus remains regarding the optimal year-end water level of a multi-year regulating storage reservoir that operates as a part of cascaded reservoirs and even mixed multi-reservoirs.

ACKNOWLEDGEMENTS

This work was supported by the Special Fund of the National Basic Research Program of China (973 Program) (No. 2013CB036406-4), and Key Program of the National Natural Science Foundation of China (No. 50539140), and the Energy Foundation of the US (China Sustainable Energy Program, No. G-0610-08581). The authors are grateful to anonymous reviewers for their useful comments and suggestions.

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